 in which \(X_1, X_2, \ldots \)  are i.i.d.  random variables and the weights \(a_1, a_2,\ldots \)  correspond to either the eigenvalues or the singular values of a random symmetric matrix. Specifically, we take eigenvalues and singular values corresponding to the Erdős-Rényi-Gilbert random graph model. A random graph in this model, which was developed independently by Erdős-Rényi [1]}, [2]} and Gilbert [3]}, is constructed by attaching edges among a set of labeled vertices independently with probability \(p\) . The random variables \(a_iX_i\)  in this case are neither independent nor indentically distributed, and there is no general method available to handle this situation. However, adjacency matrices of Erdős-Rényi-Gilbert graphs have bounded entries which, modulo the constraints imposed by symmetry, are independent. This simple fact, together, with the almost sure convergence of their empirical spectral distributions to the semicircular law, allow us to establish central limit-type theorems for the sequences \(n^{-1}Z_n(a)\) .
